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In this talk I will introduce what we call RAFH (=Rees affine Frobenius Hecke algebras) which are generalizations of and provide a common framework for several versions of type A Hecke algebras and of algebras of 2-morphisms in categorifications of type A. Then the focus of the talk is on establishing the most basic representation and structure theory in the general context. This is joint work with Andrew Mathas.